Mechanisms of Gene Regulation: Boolean Network Models of the Lactose Operon in Escherichia coli - Mathematical Concepts and Methods in Modern Biology - page 19

To analyze the network, we enter the transition functions in the Input Functions
text area as follows:

f 1 = x 4

f 2 = ( x 3 ∗ x 4 )

f 3 = ( x 2 ∗ x 4 )

f 4 = (( x 1 ∗ x 2 ) ∗ x 3 ). (1.5)

∗

DVDuses the following symbols for the basic logical operations: for AND, + for OR,

∼

and for NOT. Notice the multiple sets of parentheses. DVD requires that all Boolean

expressions are fully parenthesized , meaning that every single operation should be
enclosed in parentheses. Multiple sets of parentheses for the same operations should
not be used , as this may lead to incorrect results. In DVD the nodes are always labeled
x1, x2, and so on; their respective update rules are denoted f1, f2, and so on. If your
model uses different variable names, they need to be changed before using DVD to
conform with this requirement.
Exercise 1.9. Open DVD v1.0 and enter and run the example fromEqs. ( 1.5 ). Check
the appropriate boxes in the bottom right panel “Additional Output Specification” to
generate the state space transition graph and the dependency graph. Clicking on
the “Generate” button displays the characteristics of the Boolean network. Follow
the links to the space graph and dependency graph to examine them. Answer the
following questions: (1) How many fixed points does the model have? (2) How many
components does the state space graph have? (3) Are there any limit cycles?

Exercise 1.10. For the example from Eqs. ( 1.5 ), use DVD to find the trajectory
of the state (0, 1, 1, 1). To do so, select the radio button “One Trajectory…” in the
“State Space Specification” panel and enter the components of the space separated
by spaces: 0 1 1 1. Clicking on the “Generate” button will display the path.

DVD does not provide a specific option for designating selected nodes as param-
eters. Thus, parameter values should either be entered explicitly as 0s or 1s in the
model equations or they should be treated as “variables” that retain their constant
values for all time steps. To do the latter, for each parameter P , we add an update
rule of the form f P =

P (usually at the end of the model, after the update rules for
all of the variables). This ensures that P retains its initial value for all time steps t .
Of course, with this approach, the DVD output should be interpreted carefully and
with the understanding that different sets of initial values for the parameters should
be considered as separate outputs from the model.
Example 1.5. We will show how the lac operon model from Eqs. ( 1.4 ) can be
analyzed using DVD. First, the variable names will need to change to names accepted
by DVD. We will use x1 for M ,x2for E , and x3 for L . Two different approaches are
possible for the model parameters:

1. Enter and run the model four different times for each of the four combinations
of the parameters L e and G e (see Exercise 1.11 ). As an example, for L e = 1 and